Properties

Label 216.3.h.f.53.3
Level $216$
Weight $3$
Character 216.53
Analytic conductor $5.886$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [216,3,Mod(53,216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("216.53");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 216.h (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.88557371018\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.242095489024.11
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{6} + 2x^{4} - 32x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 53.3
Root \(-0.926315 - 1.77255i\) of defining polynomial
Character \(\chi\) \(=\) 216.53
Dual form 216.3.h.f.53.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.926315 - 1.77255i) q^{2} +(-2.28388 + 3.28388i) q^{4} -5.39773 q^{5} +11.5678 q^{7} +(7.93645 + 1.00639i) q^{8} +O(q^{10})\) \(q+(-0.926315 - 1.77255i) q^{2} +(-2.28388 + 3.28388i) q^{4} -5.39773 q^{5} +11.5678 q^{7} +(7.93645 + 1.00639i) q^{8} +(5.00000 + 9.56776i) q^{10} +1.69247 q^{11} -9.56776i q^{13} +(-10.7154 - 20.5045i) q^{14} +(-5.56776 - 15.0000i) q^{16} -19.2578i q^{17} +1.86447i q^{19} +(12.3278 - 17.7255i) q^{20} +(-1.56776 - 3.00000i) q^{22} -30.3736i q^{23} +4.13553 q^{25} +(-16.9594 + 8.86276i) q^{26} +(-26.4194 + 37.9872i) q^{28} +49.3112 q^{29} +23.4066 q^{31} +(-21.4308 + 23.7639i) q^{32} +(-34.1355 + 17.8388i) q^{34} -62.4397 q^{35} -24.9744i q^{37} +(3.30487 - 1.72709i) q^{38} +(-42.8388 - 5.43224i) q^{40} +68.7984i q^{41} -83.6776i q^{43} +(-3.86541 + 5.55789i) q^{44} +(-53.8388 + 28.1355i) q^{46} -28.2699i q^{47} +84.8132 q^{49} +(-3.83080 - 7.33044i) q^{50} +(31.4194 + 21.8516i) q^{52} +61.3879 q^{53} -9.13553 q^{55} +(91.8069 + 11.6417i) q^{56} +(-45.6776 - 87.4066i) q^{58} -75.3389 q^{59} -4.70329i q^{61} +(-21.6819 - 41.4894i) q^{62} +(61.9744 + 15.9744i) q^{64} +51.6442i q^{65} +99.8132i q^{67} +(63.2405 + 43.9826i) q^{68} +(57.8388 + 110.678i) q^{70} +73.0058i q^{71} +8.72894 q^{73} +(-44.2683 + 23.1341i) q^{74} +(-6.12270 - 4.25823i) q^{76} +19.5782 q^{77} -3.83882 q^{79} +(30.0533 + 80.9660i) q^{80} +(121.949 - 63.7289i) q^{82} +37.0526 q^{83} +103.949i q^{85} +(-148.323 + 77.5118i) q^{86} +(13.4322 + 1.70329i) q^{88} +43.4112i q^{89} -110.678i q^{91} +(99.7434 + 69.3698i) q^{92} +(-50.1099 + 26.1868i) q^{94} -10.0639i q^{95} -86.9487 q^{97} +(-78.5637 - 150.336i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 48 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} + 48 q^{7} + 40 q^{10} + 32 q^{22} - 56 q^{25} - 100 q^{28} - 80 q^{31} - 184 q^{34} - 120 q^{40} - 208 q^{46} + 144 q^{49} + 140 q^{52} + 16 q^{55} + 80 q^{58} + 184 q^{64} + 240 q^{70} + 248 q^{73} + 196 q^{76} + 192 q^{79} + 352 q^{82} + 152 q^{88} - 72 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.926315 1.77255i −0.463157 0.886276i
\(3\) 0 0
\(4\) −2.28388 + 3.28388i −0.570971 + 0.820971i
\(5\) −5.39773 −1.07955 −0.539773 0.841810i \(-0.681490\pi\)
−0.539773 + 0.841810i \(0.681490\pi\)
\(6\) 0 0
\(7\) 11.5678 1.65254 0.826269 0.563276i \(-0.190459\pi\)
0.826269 + 0.563276i \(0.190459\pi\)
\(8\) 7.93645 + 1.00639i 0.992056 + 0.125799i
\(9\) 0 0
\(10\) 5.00000 + 9.56776i 0.500000 + 0.956776i
\(11\) 1.69247 0.153861 0.0769307 0.997036i \(-0.475488\pi\)
0.0769307 + 0.997036i \(0.475488\pi\)
\(12\) 0 0
\(13\) 9.56776i 0.735982i −0.929829 0.367991i \(-0.880046\pi\)
0.929829 0.367991i \(-0.119954\pi\)
\(14\) −10.7154 20.5045i −0.765385 1.46460i
\(15\) 0 0
\(16\) −5.56776 15.0000i −0.347985 0.937500i
\(17\) 19.2578i 1.13281i −0.824126 0.566407i \(-0.808333\pi\)
0.824126 0.566407i \(-0.191667\pi\)
\(18\) 0 0
\(19\) 1.86447i 0.0981301i 0.998796 + 0.0490650i \(0.0156242\pi\)
−0.998796 + 0.0490650i \(0.984376\pi\)
\(20\) 12.3278 17.7255i 0.616389 0.886276i
\(21\) 0 0
\(22\) −1.56776 3.00000i −0.0712620 0.136364i
\(23\) 30.3736i 1.32059i −0.751005 0.660296i \(-0.770431\pi\)
0.751005 0.660296i \(-0.229569\pi\)
\(24\) 0 0
\(25\) 4.13553 0.165421
\(26\) −16.9594 + 8.86276i −0.652283 + 0.340875i
\(27\) 0 0
\(28\) −26.4194 + 37.9872i −0.943550 + 1.35668i
\(29\) 49.3112 1.70038 0.850192 0.526472i \(-0.176486\pi\)
0.850192 + 0.526472i \(0.176486\pi\)
\(30\) 0 0
\(31\) 23.4066 0.755051 0.377526 0.925999i \(-0.376775\pi\)
0.377526 + 0.925999i \(0.376775\pi\)
\(32\) −21.4308 + 23.7639i −0.669712 + 0.742621i
\(33\) 0 0
\(34\) −34.1355 + 17.8388i −1.00399 + 0.524671i
\(35\) −62.4397 −1.78399
\(36\) 0 0
\(37\) 24.9744i 0.674982i −0.941329 0.337491i \(-0.890422\pi\)
0.941329 0.337491i \(-0.109578\pi\)
\(38\) 3.30487 1.72709i 0.0869703 0.0454497i
\(39\) 0 0
\(40\) −42.8388 5.43224i −1.07097 0.135806i
\(41\) 68.7984i 1.67801i 0.544124 + 0.839005i \(0.316862\pi\)
−0.544124 + 0.839005i \(0.683138\pi\)
\(42\) 0 0
\(43\) 83.6776i 1.94599i −0.230823 0.972996i \(-0.574142\pi\)
0.230823 0.972996i \(-0.425858\pi\)
\(44\) −3.86541 + 5.55789i −0.0878503 + 0.126316i
\(45\) 0 0
\(46\) −53.8388 + 28.1355i −1.17041 + 0.611642i
\(47\) 28.2699i 0.601487i −0.953705 0.300744i \(-0.902765\pi\)
0.953705 0.300744i \(-0.0972349\pi\)
\(48\) 0 0
\(49\) 84.8132 1.73088
\(50\) −3.83080 7.33044i −0.0766160 0.146609i
\(51\) 0 0
\(52\) 31.4194 + 21.8516i 0.604219 + 0.420224i
\(53\) 61.3879 1.15826 0.579131 0.815235i \(-0.303392\pi\)
0.579131 + 0.815235i \(0.303392\pi\)
\(54\) 0 0
\(55\) −9.13553 −0.166101
\(56\) 91.8069 + 11.6417i 1.63941 + 0.207888i
\(57\) 0 0
\(58\) −45.6776 87.4066i −0.787546 1.50701i
\(59\) −75.3389 −1.27693 −0.638465 0.769651i \(-0.720430\pi\)
−0.638465 + 0.769651i \(0.720430\pi\)
\(60\) 0 0
\(61\) 4.70329i 0.0771032i −0.999257 0.0385516i \(-0.987726\pi\)
0.999257 0.0385516i \(-0.0122744\pi\)
\(62\) −21.6819 41.4894i −0.349707 0.669184i
\(63\) 0 0
\(64\) 61.9744 + 15.9744i 0.968349 + 0.249599i
\(65\) 51.6442i 0.794527i
\(66\) 0 0
\(67\) 99.8132i 1.48975i 0.667205 + 0.744874i \(0.267491\pi\)
−0.667205 + 0.744874i \(0.732509\pi\)
\(68\) 63.2405 + 43.9826i 0.930007 + 0.646804i
\(69\) 0 0
\(70\) 57.8388 + 110.678i 0.826269 + 1.58111i
\(71\) 73.0058i 1.02825i 0.857715 + 0.514125i \(0.171883\pi\)
−0.857715 + 0.514125i \(0.828117\pi\)
\(72\) 0 0
\(73\) 8.72894 0.119575 0.0597873 0.998211i \(-0.480958\pi\)
0.0597873 + 0.998211i \(0.480958\pi\)
\(74\) −44.2683 + 23.1341i −0.598221 + 0.312623i
\(75\) 0 0
\(76\) −6.12270 4.25823i −0.0805619 0.0560294i
\(77\) 19.5782 0.254262
\(78\) 0 0
\(79\) −3.83882 −0.0485927 −0.0242963 0.999705i \(-0.507735\pi\)
−0.0242963 + 0.999705i \(0.507735\pi\)
\(80\) 30.0533 + 80.9660i 0.375666 + 1.01208i
\(81\) 0 0
\(82\) 121.949 63.7289i 1.48718 0.777182i
\(83\) 37.0526 0.446417 0.223208 0.974771i \(-0.428347\pi\)
0.223208 + 0.974771i \(0.428347\pi\)
\(84\) 0 0
\(85\) 103.949i 1.22293i
\(86\) −148.323 + 77.5118i −1.72469 + 0.901300i
\(87\) 0 0
\(88\) 13.4322 + 1.70329i 0.152639 + 0.0193556i
\(89\) 43.4112i 0.487767i 0.969805 + 0.243883i \(0.0784214\pi\)
−0.969805 + 0.243883i \(0.921579\pi\)
\(90\) 0 0
\(91\) 110.678i 1.21624i
\(92\) 99.7434 + 69.3698i 1.08417 + 0.754019i
\(93\) 0 0
\(94\) −50.1099 + 26.1868i −0.533084 + 0.278583i
\(95\) 10.0639i 0.105936i
\(96\) 0 0
\(97\) −86.9487 −0.896378 −0.448189 0.893939i \(-0.647931\pi\)
−0.448189 + 0.893939i \(0.647931\pi\)
\(98\) −78.5637 150.336i −0.801670 1.53404i
\(99\) 0 0
\(100\) −9.44506 + 13.5806i −0.0944506 + 0.135806i
\(101\) 51.4149 0.509058 0.254529 0.967065i \(-0.418080\pi\)
0.254529 + 0.967065i \(0.418080\pi\)
\(102\) 0 0
\(103\) 47.2454 0.458693 0.229347 0.973345i \(-0.426341\pi\)
0.229347 + 0.973345i \(0.426341\pi\)
\(104\) 9.62892 75.9340i 0.0925858 0.730135i
\(105\) 0 0
\(106\) −56.8645 108.813i −0.536457 1.02654i
\(107\) −75.1570 −0.702402 −0.351201 0.936300i \(-0.614227\pi\)
−0.351201 + 0.936300i \(0.614227\pi\)
\(108\) 0 0
\(109\) 133.136i 1.22143i −0.791852 0.610713i \(-0.790883\pi\)
0.791852 0.610713i \(-0.209117\pi\)
\(110\) 8.46237 + 16.1932i 0.0769307 + 0.147211i
\(111\) 0 0
\(112\) −64.4066 173.516i −0.575059 1.54925i
\(113\) 90.1599i 0.797875i −0.916978 0.398938i \(-0.869379\pi\)
0.916978 0.398938i \(-0.130621\pi\)
\(114\) 0 0
\(115\) 163.949i 1.42564i
\(116\) −112.621 + 161.932i −0.970869 + 1.39597i
\(117\) 0 0
\(118\) 69.7875 + 133.542i 0.591420 + 1.13171i
\(119\) 222.770i 1.87202i
\(120\) 0 0
\(121\) −118.136 −0.976327
\(122\) −8.33683 + 4.35673i −0.0683347 + 0.0357109i
\(123\) 0 0
\(124\) −53.4579 + 76.8645i −0.431112 + 0.619875i
\(125\) 112.621 0.900967
\(126\) 0 0
\(127\) −111.355 −0.876813 −0.438407 0.898777i \(-0.644457\pi\)
−0.438407 + 0.898777i \(0.644457\pi\)
\(128\) −29.0924 124.650i −0.227284 0.973828i
\(129\) 0 0
\(130\) 91.5421 47.8388i 0.704170 0.367991i
\(131\) −92.4930 −0.706054 −0.353027 0.935613i \(-0.614848\pi\)
−0.353027 + 0.935613i \(0.614848\pi\)
\(132\) 0 0
\(133\) 21.5678i 0.162164i
\(134\) 176.924 92.4584i 1.32033 0.689988i
\(135\) 0 0
\(136\) 19.3809 152.839i 0.142507 1.12381i
\(137\) 199.396i 1.45544i 0.685872 + 0.727722i \(0.259421\pi\)
−0.685872 + 0.727722i \(0.740579\pi\)
\(138\) 0 0
\(139\) 99.4908i 0.715761i 0.933767 + 0.357881i \(0.116501\pi\)
−0.933767 + 0.357881i \(0.883499\pi\)
\(140\) 142.605 205.045i 1.01861 1.46460i
\(141\) 0 0
\(142\) 129.407 67.6263i 0.911314 0.476242i
\(143\) 16.1932i 0.113239i
\(144\) 0 0
\(145\) −266.168 −1.83564
\(146\) −8.08575 15.4725i −0.0553818 0.105976i
\(147\) 0 0
\(148\) 82.0128 + 57.0385i 0.554141 + 0.385395i
\(149\) 100.085 0.671714 0.335857 0.941913i \(-0.390974\pi\)
0.335857 + 0.941913i \(0.390974\pi\)
\(150\) 0 0
\(151\) −185.788 −1.23038 −0.615190 0.788379i \(-0.710921\pi\)
−0.615190 + 0.788379i \(0.710921\pi\)
\(152\) −1.87639 + 14.7973i −0.0123447 + 0.0973505i
\(153\) 0 0
\(154\) −18.1355 34.7033i −0.117763 0.225346i
\(155\) −126.343 −0.815113
\(156\) 0 0
\(157\) 201.575i 1.28392i 0.766739 + 0.641959i \(0.221878\pi\)
−0.766739 + 0.641959i \(0.778122\pi\)
\(158\) 3.55596 + 6.80451i 0.0225061 + 0.0430665i
\(159\) 0 0
\(160\) 115.678 128.271i 0.722985 0.801694i
\(161\) 351.355i 2.18233i
\(162\) 0 0
\(163\) 19.8645i 0.121868i −0.998142 0.0609340i \(-0.980592\pi\)
0.998142 0.0609340i \(-0.0194079\pi\)
\(164\) −225.926 157.127i −1.37760 0.958094i
\(165\) 0 0
\(166\) −34.3224 65.6776i −0.206761 0.395648i
\(167\) 22.3225i 0.133668i −0.997764 0.0668338i \(-0.978710\pi\)
0.997764 0.0668338i \(-0.0212897\pi\)
\(168\) 0 0
\(169\) 77.4579 0.458331
\(170\) 184.254 96.2892i 1.08385 0.566407i
\(171\) 0 0
\(172\) 274.788 + 191.110i 1.59760 + 1.11110i
\(173\) −262.476 −1.51720 −0.758602 0.651555i \(-0.774117\pi\)
−0.758602 + 0.651555i \(0.774117\pi\)
\(174\) 0 0
\(175\) 47.8388 0.273365
\(176\) −9.42330 25.3871i −0.0535415 0.144245i
\(177\) 0 0
\(178\) 76.9487 40.2125i 0.432296 0.225913i
\(179\) −88.1037 −0.492200 −0.246100 0.969244i \(-0.579149\pi\)
−0.246100 + 0.969244i \(0.579149\pi\)
\(180\) 0 0
\(181\) 201.568i 1.11363i 0.830635 + 0.556817i \(0.187978\pi\)
−0.830635 + 0.556817i \(0.812022\pi\)
\(182\) −196.182 + 102.522i −1.07792 + 0.563309i
\(183\) 0 0
\(184\) 30.5678 241.059i 0.166129 1.31010i
\(185\) 134.805i 0.728675i
\(186\) 0 0
\(187\) 32.5934i 0.174296i
\(188\) 92.8350 + 64.5651i 0.493803 + 0.343432i
\(189\) 0 0
\(190\) −17.8388 + 9.32236i −0.0938885 + 0.0490650i
\(191\) 28.0880i 0.147058i 0.997293 + 0.0735289i \(0.0234261\pi\)
−0.997293 + 0.0735289i \(0.976574\pi\)
\(192\) 0 0
\(193\) 265.762 1.37700 0.688502 0.725234i \(-0.258269\pi\)
0.688502 + 0.725234i \(0.258269\pi\)
\(194\) 80.5419 + 154.121i 0.415164 + 0.794439i
\(195\) 0 0
\(196\) −193.703 + 278.516i −0.988282 + 1.42100i
\(197\) −27.9930 −0.142096 −0.0710482 0.997473i \(-0.522634\pi\)
−0.0710482 + 0.997473i \(0.522634\pi\)
\(198\) 0 0
\(199\) 7.34801 0.0369247 0.0184623 0.999830i \(-0.494123\pi\)
0.0184623 + 0.999830i \(0.494123\pi\)
\(200\) 32.8214 + 4.16196i 0.164107 + 0.0208098i
\(201\) 0 0
\(202\) −47.6263 91.1355i −0.235774 0.451166i
\(203\) 570.420 2.80995
\(204\) 0 0
\(205\) 371.355i 1.81149i
\(206\) −43.7641 83.7449i −0.212447 0.406529i
\(207\) 0 0
\(208\) −143.516 + 53.2711i −0.689983 + 0.256111i
\(209\) 3.15557i 0.0150984i
\(210\) 0 0
\(211\) 140.678i 0.666719i −0.942800 0.333359i \(-0.891818\pi\)
0.942800 0.333359i \(-0.108182\pi\)
\(212\) −140.203 + 201.590i −0.661333 + 0.950898i
\(213\) 0 0
\(214\) 69.6191 + 133.220i 0.325323 + 0.622522i
\(215\) 451.670i 2.10079i
\(216\) 0 0
\(217\) 270.762 1.24775
\(218\) −235.990 + 123.325i −1.08252 + 0.565713i
\(219\) 0 0
\(220\) 20.8645 30.0000i 0.0948385 0.136364i
\(221\) −184.254 −0.833731
\(222\) 0 0
\(223\) −0.593414 −0.00266105 −0.00133052 0.999999i \(-0.500424\pi\)
−0.00133052 + 0.999999i \(0.500424\pi\)
\(224\) −247.906 + 274.895i −1.10672 + 1.22721i
\(225\) 0 0
\(226\) −159.813 + 83.5165i −0.707138 + 0.369542i
\(227\) 381.044 1.67861 0.839305 0.543661i \(-0.182962\pi\)
0.839305 + 0.543661i \(0.182962\pi\)
\(228\) 0 0
\(229\) 127.458i 0.556585i 0.960496 + 0.278292i \(0.0897684\pi\)
−0.960496 + 0.278292i \(0.910232\pi\)
\(230\) 290.608 151.868i 1.26351 0.660296i
\(231\) 0 0
\(232\) 391.355 + 49.6263i 1.68688 + 0.213907i
\(233\) 427.153i 1.83327i −0.399722 0.916636i \(-0.630893\pi\)
0.399722 0.916636i \(-0.369107\pi\)
\(234\) 0 0
\(235\) 152.593i 0.649334i
\(236\) 172.065 247.404i 0.729090 1.04832i
\(237\) 0 0
\(238\) −394.872 + 206.355i −1.65913 + 0.867039i
\(239\) 190.475i 0.796965i −0.917176 0.398483i \(-0.869537\pi\)
0.917176 0.398483i \(-0.130463\pi\)
\(240\) 0 0
\(241\) −2.78024 −0.0115363 −0.00576814 0.999983i \(-0.501836\pi\)
−0.00576814 + 0.999983i \(0.501836\pi\)
\(242\) 109.431 + 209.401i 0.452193 + 0.865295i
\(243\) 0 0
\(244\) 15.4451 + 10.7418i 0.0632994 + 0.0440236i
\(245\) −457.799 −1.86857
\(246\) 0 0
\(247\) 17.8388 0.0722220
\(248\) 185.765 + 23.5562i 0.749053 + 0.0949847i
\(249\) 0 0
\(250\) −104.322 199.626i −0.417289 0.798505i
\(251\) 100.085 0.398747 0.199373 0.979924i \(-0.436109\pi\)
0.199373 + 0.979924i \(0.436109\pi\)
\(252\) 0 0
\(253\) 51.4066i 0.203188i
\(254\) 103.150 + 197.383i 0.406103 + 0.777099i
\(255\) 0 0
\(256\) −194.000 + 167.033i −0.757812 + 0.652472i
\(257\) 344.719i 1.34132i 0.741765 + 0.670660i \(0.233989\pi\)
−0.741765 + 0.670660i \(0.766011\pi\)
\(258\) 0 0
\(259\) 288.897i 1.11543i
\(260\) −169.594 117.949i −0.652283 0.453651i
\(261\) 0 0
\(262\) 85.6776 + 163.949i 0.327014 + 0.625758i
\(263\) 169.801i 0.645632i 0.946462 + 0.322816i \(0.104630\pi\)
−0.946462 + 0.322816i \(0.895370\pi\)
\(264\) 0 0
\(265\) −331.355 −1.25040
\(266\) 38.2300 19.9785i 0.143722 0.0751073i
\(267\) 0 0
\(268\) −327.775 227.962i −1.22304 0.850603i
\(269\) 263.666 0.980173 0.490086 0.871674i \(-0.336965\pi\)
0.490086 + 0.871674i \(0.336965\pi\)
\(270\) 0 0
\(271\) 55.6849 0.205479 0.102740 0.994708i \(-0.467239\pi\)
0.102740 + 0.994708i \(0.467239\pi\)
\(272\) −288.868 + 107.223i −1.06201 + 0.394203i
\(273\) 0 0
\(274\) 353.440 184.703i 1.28993 0.674100i
\(275\) 6.99928 0.0254519
\(276\) 0 0
\(277\) 244.051i 0.881052i 0.897740 + 0.440526i \(0.145208\pi\)
−0.897740 + 0.440526i \(0.854792\pi\)
\(278\) 176.353 92.1598i 0.634362 0.331510i
\(279\) 0 0
\(280\) −495.549 62.8388i −1.76982 0.224424i
\(281\) 123.780i 0.440498i 0.975444 + 0.220249i \(0.0706871\pi\)
−0.975444 + 0.220249i \(0.929313\pi\)
\(282\) 0 0
\(283\) 41.5092i 0.146676i 0.997307 + 0.0733378i \(0.0233651\pi\)
−0.997307 + 0.0733378i \(0.976635\pi\)
\(284\) −239.742 166.737i −0.844164 0.587101i
\(285\) 0 0
\(286\) −28.7033 + 15.0000i −0.100361 + 0.0524476i
\(287\) 795.843i 2.77297i
\(288\) 0 0
\(289\) −81.8645 −0.283268
\(290\) 246.556 + 471.797i 0.850192 + 1.62689i
\(291\) 0 0
\(292\) −19.9359 + 28.6648i −0.0682735 + 0.0981672i
\(293\) −371.985 −1.26957 −0.634786 0.772688i \(-0.718912\pi\)
−0.634786 + 0.772688i \(0.718912\pi\)
\(294\) 0 0
\(295\) 406.659 1.37851
\(296\) 25.1340 198.208i 0.0849121 0.669620i
\(297\) 0 0
\(298\) −92.7106 177.407i −0.311109 0.595324i
\(299\) −290.608 −0.971932
\(300\) 0 0
\(301\) 967.963i 3.21582i
\(302\) 172.098 + 329.318i 0.569860 + 1.09046i
\(303\) 0 0
\(304\) 27.9671 10.3809i 0.0919969 0.0341478i
\(305\) 25.3871i 0.0832365i
\(306\) 0 0
\(307\) 215.033i 0.700433i −0.936669 0.350216i \(-0.886108\pi\)
0.936669 0.350216i \(-0.113892\pi\)
\(308\) −44.7142 + 64.2923i −0.145176 + 0.208741i
\(309\) 0 0
\(310\) 117.033 + 223.949i 0.377526 + 0.722415i
\(311\) 200.539i 0.644819i −0.946600 0.322409i \(-0.895507\pi\)
0.946600 0.322409i \(-0.104493\pi\)
\(312\) 0 0
\(313\) 142.033 0.453779 0.226890 0.973920i \(-0.427144\pi\)
0.226890 + 0.973920i \(0.427144\pi\)
\(314\) 357.302 186.722i 1.13791 0.594656i
\(315\) 0 0
\(316\) 8.76742 12.6062i 0.0277450 0.0398932i
\(317\) −3.20309 −0.0101044 −0.00505219 0.999987i \(-0.501608\pi\)
−0.00505219 + 0.999987i \(0.501608\pi\)
\(318\) 0 0
\(319\) 83.4579 0.261623
\(320\) −334.521 86.2253i −1.04538 0.269454i
\(321\) 0 0
\(322\) −622.795 + 325.465i −1.93415 + 1.01076i
\(323\) 35.9057 0.111163
\(324\) 0 0
\(325\) 39.5678i 0.121747i
\(326\) −35.2108 + 18.4008i −0.108009 + 0.0564440i
\(327\) 0 0
\(328\) −69.2381 + 546.015i −0.211092 + 1.66468i
\(329\) 327.020i 0.993981i
\(330\) 0 0
\(331\) 328.136i 0.991346i 0.868509 + 0.495673i \(0.165079\pi\)
−0.868509 + 0.495673i \(0.834921\pi\)
\(332\) −84.6237 + 121.676i −0.254891 + 0.366495i
\(333\) 0 0
\(334\) −39.5678 + 20.6776i −0.118466 + 0.0619091i
\(335\) 538.765i 1.60825i
\(336\) 0 0
\(337\) 430.678 1.27798 0.638988 0.769217i \(-0.279354\pi\)
0.638988 + 0.769217i \(0.279354\pi\)
\(338\) −71.7504 137.298i −0.212279 0.406208i
\(339\) 0 0
\(340\) −341.355 237.407i −1.00399 0.698255i
\(341\) 39.6151 0.116173
\(342\) 0 0
\(343\) 414.278 1.20781
\(344\) 84.2125 664.103i 0.244804 1.93053i
\(345\) 0 0
\(346\) 243.136 + 465.253i 0.702704 + 1.34466i
\(347\) 152.418 0.439244 0.219622 0.975585i \(-0.429518\pi\)
0.219622 + 0.975585i \(0.429518\pi\)
\(348\) 0 0
\(349\) 370.718i 1.06223i 0.847300 + 0.531114i \(0.178227\pi\)
−0.847300 + 0.531114i \(0.821773\pi\)
\(350\) −44.3138 84.7968i −0.126611 0.242277i
\(351\) 0 0
\(352\) −36.2711 + 40.2198i −0.103043 + 0.114261i
\(353\) 64.7728i 0.183492i −0.995782 0.0917462i \(-0.970755\pi\)
0.995782 0.0917462i \(-0.0292448\pi\)
\(354\) 0 0
\(355\) 394.066i 1.11004i
\(356\) −142.557 99.1462i −0.400442 0.278500i
\(357\) 0 0
\(358\) 81.6118 + 156.168i 0.227966 + 0.436225i
\(359\) 375.093i 1.04483i −0.852692 0.522414i \(-0.825032\pi\)
0.852692 0.522414i \(-0.174968\pi\)
\(360\) 0 0
\(361\) 357.524 0.990370
\(362\) 357.289 186.715i 0.986987 0.515788i
\(363\) 0 0
\(364\) 363.452 + 252.775i 0.998495 + 0.694436i
\(365\) −47.1165 −0.129086
\(366\) 0 0
\(367\) 367.839 1.00229 0.501143 0.865365i \(-0.332913\pi\)
0.501143 + 0.865365i \(0.332913\pi\)
\(368\) −455.604 + 169.113i −1.23806 + 0.459547i
\(369\) 0 0
\(370\) 238.949 124.872i 0.645807 0.337491i
\(371\) 710.120 1.91407
\(372\) 0 0
\(373\) 247.128i 0.662542i 0.943536 + 0.331271i \(0.107477\pi\)
−0.943536 + 0.331271i \(0.892523\pi\)
\(374\) −57.7735 + 30.1918i −0.154475 + 0.0807266i
\(375\) 0 0
\(376\) 28.4506 224.363i 0.0756665 0.596709i
\(377\) 471.797i 1.25145i
\(378\) 0 0
\(379\) 401.776i 1.06010i 0.847968 + 0.530048i \(0.177826\pi\)
−0.847968 + 0.530048i \(0.822174\pi\)
\(380\) 33.0487 + 22.9848i 0.0869703 + 0.0604863i
\(381\) 0 0
\(382\) 49.7875 26.0184i 0.130334 0.0681109i
\(383\) 319.475i 0.834138i 0.908875 + 0.417069i \(0.136943\pi\)
−0.908875 + 0.417069i \(0.863057\pi\)
\(384\) 0 0
\(385\) −105.678 −0.274487
\(386\) −246.179 471.077i −0.637770 1.22041i
\(387\) 0 0
\(388\) 198.581 285.529i 0.511806 0.735900i
\(389\) 213.802 0.549618 0.274809 0.961499i \(-0.411385\pi\)
0.274809 + 0.961499i \(0.411385\pi\)
\(390\) 0 0
\(391\) −584.930 −1.49599
\(392\) 673.115 + 85.3553i 1.71713 + 0.217743i
\(393\) 0 0
\(394\) 25.9303 + 49.6191i 0.0658130 + 0.125937i
\(395\) 20.7209 0.0524581
\(396\) 0 0
\(397\) 421.136i 1.06079i −0.847749 0.530397i \(-0.822043\pi\)
0.847749 0.530397i \(-0.177957\pi\)
\(398\) −6.80657 13.0247i −0.0171019 0.0327254i
\(399\) 0 0
\(400\) −23.0256 62.0329i −0.0575641 0.155082i
\(401\) 203.097i 0.506476i −0.967404 0.253238i \(-0.918504\pi\)
0.967404 0.253238i \(-0.0814956\pi\)
\(402\) 0 0
\(403\) 223.949i 0.555704i
\(404\) −117.425 + 168.840i −0.290657 + 0.417922i
\(405\) 0 0
\(406\) −528.388 1011.10i −1.30145 2.49039i
\(407\) 42.2685i 0.103854i
\(408\) 0 0
\(409\) −622.267 −1.52144 −0.760718 0.649083i \(-0.775153\pi\)
−0.760718 + 0.649083i \(0.775153\pi\)
\(410\) −658.247 + 343.992i −1.60548 + 0.839005i
\(411\) 0 0
\(412\) −107.903 + 155.148i −0.261900 + 0.376574i
\(413\) −871.503 −2.11018
\(414\) 0 0
\(415\) −200.000 −0.481928
\(416\) 227.367 + 205.045i 0.546556 + 0.492896i
\(417\) 0 0
\(418\) 5.59341 2.92305i 0.0133814 0.00699295i
\(419\) −271.397 −0.647726 −0.323863 0.946104i \(-0.604982\pi\)
−0.323863 + 0.946104i \(0.604982\pi\)
\(420\) 0 0
\(421\) 610.381i 1.44984i 0.688835 + 0.724918i \(0.258122\pi\)
−0.688835 + 0.724918i \(0.741878\pi\)
\(422\) −249.358 + 130.312i −0.590897 + 0.308796i
\(423\) 0 0
\(424\) 487.201 + 61.7802i 1.14906 + 0.145708i
\(425\) 79.6414i 0.187391i
\(426\) 0 0
\(427\) 54.4066i 0.127416i
\(428\) 171.650 246.807i 0.401051 0.576651i
\(429\) 0 0
\(430\) 800.608 418.388i 1.86188 0.972996i
\(431\) 303.282i 0.703669i −0.936062 0.351835i \(-0.885558\pi\)
0.936062 0.351835i \(-0.114442\pi\)
\(432\) 0 0
\(433\) 224.659 0.518844 0.259422 0.965764i \(-0.416468\pi\)
0.259422 + 0.965764i \(0.416468\pi\)
\(434\) −250.811 479.940i −0.577905 1.10585i
\(435\) 0 0
\(436\) 437.201 + 304.066i 1.00276 + 0.697399i
\(437\) 56.6307 0.129590
\(438\) 0 0
\(439\) −828.930 −1.88822 −0.944112 0.329625i \(-0.893078\pi\)
−0.944112 + 0.329625i \(0.893078\pi\)
\(440\) −72.5036 9.19392i −0.164781 0.0208953i
\(441\) 0 0
\(442\) 170.678 + 326.601i 0.386149 + 0.738916i
\(443\) 286.716 0.647215 0.323608 0.946191i \(-0.395104\pi\)
0.323608 + 0.946191i \(0.395104\pi\)
\(444\) 0 0
\(445\) 234.322i 0.526567i
\(446\) 0.549688 + 1.05186i 0.00123248 + 0.00235842i
\(447\) 0 0
\(448\) 716.905 + 184.788i 1.60023 + 0.412472i
\(449\) 196.928i 0.438593i 0.975658 + 0.219297i \(0.0703763\pi\)
−0.975658 + 0.219297i \(0.929624\pi\)
\(450\) 0 0
\(451\) 116.440i 0.258181i
\(452\) 296.075 + 205.915i 0.655032 + 0.455563i
\(453\) 0 0
\(454\) −352.967 675.421i −0.777461 1.48771i
\(455\) 597.408i 1.31299i
\(456\) 0 0
\(457\) 7.03293 0.0153893 0.00769467 0.999970i \(-0.497551\pi\)
0.00769467 + 0.999970i \(0.497551\pi\)
\(458\) 225.926 118.066i 0.493288 0.257786i
\(459\) 0 0
\(460\) −538.388 374.440i −1.17041 0.813999i
\(461\) 217.282 0.471326 0.235663 0.971835i \(-0.424274\pi\)
0.235663 + 0.971835i \(0.424274\pi\)
\(462\) 0 0
\(463\) −538.344 −1.16273 −0.581365 0.813643i \(-0.697481\pi\)
−0.581365 + 0.813643i \(0.697481\pi\)
\(464\) −274.553 739.667i −0.591709 1.59411i
\(465\) 0 0
\(466\) −757.150 + 395.678i −1.62479 + 0.849094i
\(467\) −612.637 −1.31186 −0.655928 0.754824i \(-0.727722\pi\)
−0.655928 + 0.754824i \(0.727722\pi\)
\(468\) 0 0
\(469\) 1154.62i 2.46187i
\(470\) 270.480 141.350i 0.575489 0.300744i
\(471\) 0 0
\(472\) −597.923 75.8205i −1.26679 0.160637i
\(473\) 141.622i 0.299413i
\(474\) 0 0
\(475\) 7.71057i 0.0162328i
\(476\) 731.551 + 508.781i 1.53687 + 1.06887i
\(477\) 0 0
\(478\) −337.626 + 176.440i −0.706331 + 0.369120i
\(479\) 103.652i 0.216393i 0.994130 + 0.108196i \(0.0345076\pi\)
−0.994130 + 0.108196i \(0.965492\pi\)
\(480\) 0 0
\(481\) −238.949 −0.496775
\(482\) 2.57538 + 4.92812i 0.00534311 + 0.0102243i
\(483\) 0 0
\(484\) 269.808 387.943i 0.557454 0.801535i
\(485\) 469.326 0.967682
\(486\) 0 0
\(487\) 163.516 0.335763 0.167881 0.985807i \(-0.446307\pi\)
0.167881 + 0.985807i \(0.446307\pi\)
\(488\) 4.73336 37.3274i 0.00969950 0.0764906i
\(489\) 0 0
\(490\) 424.066 + 811.472i 0.865441 + 1.65607i
\(491\) −770.543 −1.56933 −0.784667 0.619917i \(-0.787166\pi\)
−0.784667 + 0.619917i \(0.787166\pi\)
\(492\) 0 0
\(493\) 949.626i 1.92622i
\(494\) −16.5244 31.6202i −0.0334501 0.0640086i
\(495\) 0 0
\(496\) −130.322 351.099i −0.262747 0.707860i
\(497\) 844.514i 1.69922i
\(498\) 0 0
\(499\) 345.253i 0.691889i 0.938255 + 0.345945i \(0.112441\pi\)
−0.938255 + 0.345945i \(0.887559\pi\)
\(500\) −257.213 + 369.834i −0.514426 + 0.739667i
\(501\) 0 0
\(502\) −92.7106 177.407i −0.184682 0.353400i
\(503\) 358.445i 0.712614i −0.934369 0.356307i \(-0.884036\pi\)
0.934369 0.356307i \(-0.115964\pi\)
\(504\) 0 0
\(505\) −277.524 −0.549552
\(506\) −91.1209 + 47.6187i −0.180081 + 0.0941081i
\(507\) 0 0
\(508\) 254.322 365.678i 0.500635 0.719838i
\(509\) 403.367 0.792469 0.396235 0.918149i \(-0.370317\pi\)
0.396235 + 0.918149i \(0.370317\pi\)
\(510\) 0 0
\(511\) 100.974 0.197601
\(512\) 475.780 + 189.150i 0.929257 + 0.369434i
\(513\) 0 0
\(514\) 611.033 319.319i 1.18878 0.621242i
\(515\) −255.018 −0.495181
\(516\) 0 0
\(517\) 47.8461i 0.0925457i
\(518\) −512.086 + 267.610i −0.988582 + 0.516621i
\(519\) 0 0
\(520\) −51.9744 + 409.872i −0.0999507 + 0.788215i
\(521\) 835.047i 1.60278i 0.598144 + 0.801389i \(0.295905\pi\)
−0.598144 + 0.801389i \(0.704095\pi\)
\(522\) 0 0
\(523\) 21.1539i 0.0404472i 0.999795 + 0.0202236i \(0.00643781\pi\)
−0.999795 + 0.0202236i \(0.993562\pi\)
\(524\) 211.243 303.736i 0.403136 0.579649i
\(525\) 0 0
\(526\) 300.982 157.289i 0.572208 0.299029i
\(527\) 450.760i 0.855333i
\(528\) 0 0
\(529\) −393.557 −0.743963
\(530\) 306.939 + 587.345i 0.579131 + 1.10820i
\(531\) 0 0
\(532\) −70.8260 49.2582i −0.133132 0.0925907i
\(533\) 658.247 1.23498
\(534\) 0 0
\(535\) 405.678 0.758276
\(536\) −100.451 + 792.162i −0.187409 + 1.47791i
\(537\) 0 0
\(538\) −244.238 467.363i −0.453974 0.868704i
\(539\) 143.544 0.266316
\(540\) 0 0
\(541\) 130.718i 0.241623i 0.992675 + 0.120811i \(0.0385496\pi\)
−0.992675 + 0.120811i \(0.961450\pi\)
\(542\) −51.5818 98.7044i −0.0951693 0.182112i
\(543\) 0 0
\(544\) 457.641 + 412.711i 0.841252 + 0.758659i
\(545\) 718.630i 1.31859i
\(546\) 0 0
\(547\) 544.963i 0.996277i 0.867098 + 0.498138i \(0.165983\pi\)
−0.867098 + 0.498138i \(0.834017\pi\)
\(548\) −654.792 455.397i −1.19488 0.831016i
\(549\) 0 0
\(550\) −6.48353 12.4066i −0.0117882 0.0225574i
\(551\) 91.9392i 0.166859i
\(552\) 0 0
\(553\) −44.4066 −0.0803012
\(554\) 432.594 226.068i 0.780855 0.408066i
\(555\) 0 0
\(556\) −326.716 227.225i −0.587619 0.408679i
\(557\) 796.393 1.42979 0.714895 0.699232i \(-0.246474\pi\)
0.714895 + 0.699232i \(0.246474\pi\)
\(558\) 0 0
\(559\) −800.608 −1.43221
\(560\) 347.650 + 936.596i 0.620803 + 1.67249i
\(561\) 0 0
\(562\) 219.407 114.659i 0.390403 0.204020i
\(563\) −519.646 −0.922994 −0.461497 0.887142i \(-0.652687\pi\)
−0.461497 + 0.887142i \(0.652687\pi\)
\(564\) 0 0
\(565\) 486.659i 0.861344i
\(566\) 73.5772 38.4506i 0.129995 0.0679339i
\(567\) 0 0
\(568\) −73.4724 + 579.407i −0.129353 + 1.02008i
\(569\) 676.128i 1.18827i −0.804364 0.594137i \(-0.797494\pi\)
0.804364 0.594137i \(-0.202506\pi\)
\(570\) 0 0
\(571\) 1048.64i 1.83650i −0.396002 0.918249i \(-0.629603\pi\)
0.396002 0.918249i \(-0.370397\pi\)
\(572\) 53.1766 + 36.9834i 0.0929660 + 0.0646562i
\(573\) 0 0
\(574\) 1410.67 737.201i 2.45762 1.28432i
\(575\) 125.611i 0.218454i
\(576\) 0 0
\(577\) −494.150 −0.856413 −0.428206 0.903681i \(-0.640854\pi\)
−0.428206 + 0.903681i \(0.640854\pi\)
\(578\) 75.8323 + 145.109i 0.131198 + 0.251054i
\(579\) 0 0
\(580\) 607.897 874.066i 1.04810 1.50701i
\(581\) 428.616 0.737720
\(582\) 0 0
\(583\) 103.897 0.178212
\(584\) 69.2768 + 8.78474i 0.118625 + 0.0150424i
\(585\) 0 0
\(586\) 344.575 + 659.363i 0.588012 + 1.12519i
\(587\) −737.611 −1.25658 −0.628289 0.777980i \(-0.716244\pi\)
−0.628289 + 0.777980i \(0.716244\pi\)
\(588\) 0 0
\(589\) 43.6409i 0.0740932i
\(590\) −376.694 720.825i −0.638465 1.22174i
\(591\) 0 0
\(592\) −374.615 + 139.051i −0.632796 + 0.234884i
\(593\) 528.883i 0.891877i 0.895064 + 0.445938i \(0.147130\pi\)
−0.895064 + 0.445938i \(0.852870\pi\)
\(594\) 0 0
\(595\) 1202.45i 2.02093i
\(596\) −228.583 + 328.669i −0.383529 + 0.551457i
\(597\) 0 0
\(598\) 269.194 + 515.117i 0.450157 + 0.861400i
\(599\) 576.826i 0.962982i −0.876451 0.481491i \(-0.840095\pi\)
0.876451 0.481491i \(-0.159905\pi\)
\(600\) 0 0
\(601\) 353.407 0.588031 0.294015 0.955801i \(-0.405008\pi\)
0.294015 + 0.955801i \(0.405008\pi\)
\(602\) −1715.77 + 896.639i −2.85011 + 1.48943i
\(603\) 0 0
\(604\) 424.317 610.104i 0.702511 1.01011i
\(605\) 637.664 1.05399
\(606\) 0 0
\(607\) 585.634 0.964800 0.482400 0.875951i \(-0.339765\pi\)
0.482400 + 0.875951i \(0.339765\pi\)
\(608\) −44.3071 39.9571i −0.0728735 0.0657189i
\(609\) 0 0
\(610\) 45.0000 23.5165i 0.0737705 0.0385516i
\(611\) −270.480 −0.442684
\(612\) 0 0
\(613\) 301.282i 0.491488i 0.969335 + 0.245744i \(0.0790323\pi\)
−0.969335 + 0.245744i \(0.920968\pi\)
\(614\) −381.157 + 199.188i −0.620777 + 0.324411i
\(615\) 0 0
\(616\) 155.381 + 19.7033i 0.252242 + 0.0319859i
\(617\) 203.058i 0.329105i 0.986368 + 0.164552i \(0.0526180\pi\)
−0.986368 + 0.164552i \(0.947382\pi\)
\(618\) 0 0
\(619\) 667.879i 1.07896i 0.841997 + 0.539482i \(0.181380\pi\)
−0.841997 + 0.539482i \(0.818620\pi\)
\(620\) 288.551 414.894i 0.465406 0.669184i
\(621\) 0 0
\(622\) −355.465 + 185.762i −0.571487 + 0.298653i
\(623\) 502.171i 0.806053i
\(624\) 0 0
\(625\) −711.286 −1.13806
\(626\) −131.567 251.761i −0.210171 0.402174i
\(627\) 0 0
\(628\) −661.949 460.374i −1.05406 0.733079i
\(629\) −480.952 −0.764630
\(630\) 0 0
\(631\) 874.755 1.38630 0.693149 0.720794i \(-0.256223\pi\)
0.693149 + 0.720794i \(0.256223\pi\)
\(632\) −30.4666 3.86336i −0.0482066 0.00611291i
\(633\) 0 0
\(634\) 2.96707 + 5.67764i 0.00467992 + 0.00895527i
\(635\) 601.066 0.946561
\(636\) 0 0
\(637\) 811.472i 1.27390i
\(638\) −77.3083 147.933i −0.121173 0.231871i
\(639\) 0 0
\(640\) 157.033 + 672.828i 0.245364 + 1.05129i
\(641\) 160.738i 0.250761i −0.992109 0.125380i \(-0.959985\pi\)
0.992109 0.125380i \(-0.0400151\pi\)
\(642\) 0 0
\(643\) 133.949i 0.208318i 0.994561 + 0.104159i \(0.0332151\pi\)
−0.994561 + 0.104159i \(0.966785\pi\)
\(644\) 1153.81 + 802.453i 1.79163 + 1.24605i
\(645\) 0 0
\(646\) −33.2600 63.6447i −0.0514860 0.0985212i
\(647\) 383.417i 0.592607i 0.955094 + 0.296304i \(0.0957540\pi\)
−0.955094 + 0.296304i \(0.904246\pi\)
\(648\) 0 0
\(649\) −127.509 −0.196470
\(650\) −70.1359 + 36.6522i −0.107901 + 0.0563880i
\(651\) 0 0
\(652\) 65.2326 + 45.3681i 0.100050 + 0.0695830i
\(653\) −367.323 −0.562516 −0.281258 0.959632i \(-0.590752\pi\)
−0.281258 + 0.959632i \(0.590752\pi\)
\(654\) 0 0
\(655\) 499.253 0.762218
\(656\) 1031.98 383.053i 1.57313 0.583922i
\(657\) 0 0
\(658\) −579.659 + 302.923i −0.880941 + 0.460369i
\(659\) −584.418 −0.886826 −0.443413 0.896317i \(-0.646232\pi\)
−0.443413 + 0.896317i \(0.646232\pi\)
\(660\) 0 0
\(661\) 526.059i 0.795853i 0.917417 + 0.397926i \(0.130270\pi\)
−0.917417 + 0.397926i \(0.869730\pi\)
\(662\) 581.637 303.957i 0.878606 0.459149i
\(663\) 0 0
\(664\) 294.066 + 37.2894i 0.442870 + 0.0561588i
\(665\) 116.417i 0.175063i
\(666\) 0 0
\(667\) 1497.76i 2.24551i
\(668\) 73.3044 + 50.9819i 0.109737 + 0.0763202i
\(669\) 0 0
\(670\) −954.989 + 499.066i −1.42536 + 0.744874i
\(671\) 7.96021i 0.0118632i
\(672\) 0 0
\(673\) 952.538 1.41536 0.707681 0.706532i \(-0.249742\pi\)
0.707681 + 0.706532i \(0.249742\pi\)
\(674\) −398.943 763.399i −0.591904 1.13264i
\(675\) 0 0
\(676\) −176.905 + 254.363i −0.261693 + 0.376276i
\(677\) −425.772 −0.628910 −0.314455 0.949272i \(-0.601822\pi\)
−0.314455 + 0.949272i \(0.601822\pi\)
\(678\) 0 0
\(679\) −1005.80 −1.48130
\(680\) −104.613 + 824.983i −0.153843 + 1.21321i
\(681\) 0 0
\(682\) −36.6960 70.2198i −0.0538065 0.102962i
\(683\) 319.752 0.468158 0.234079 0.972218i \(-0.424793\pi\)
0.234079 + 0.972218i \(0.424793\pi\)
\(684\) 0 0
\(685\) 1076.29i 1.57122i
\(686\) −383.752 734.330i −0.559405 1.07045i
\(687\) 0 0
\(688\) −1255.16 + 465.897i −1.82437 + 0.677176i
\(689\) 587.345i 0.852459i
\(690\) 0 0
\(691\) 152.454i 0.220628i −0.993897 0.110314i \(-0.964814\pi\)
0.993897 0.110314i \(-0.0351857\pi\)
\(692\) 599.465 861.941i 0.866278 1.24558i
\(693\) 0 0
\(694\) −141.187 270.168i −0.203439 0.389292i
\(695\) 537.025i 0.772698i
\(696\) 0 0
\(697\) 1324.91 1.90087
\(698\) 657.117 343.401i 0.941428 0.491979i
\(699\) 0 0
\(700\) −109.258 + 157.097i −0.156083 + 0.224424i
\(701\) 376.643 0.537294 0.268647 0.963239i \(-0.413424\pi\)
0.268647 + 0.963239i \(0.413424\pi\)
\(702\) 0 0
\(703\) 46.5640 0.0662361
\(704\) 104.890 + 27.0362i 0.148992 + 0.0384037i
\(705\) 0 0
\(706\) −114.813 + 60.0000i −0.162625 + 0.0849858i
\(707\) 594.755 0.841238
\(708\) 0 0
\(709\) 3.09151i 0.00436038i −0.999998 0.00218019i \(-0.999306\pi\)
0.999998 0.00218019i \(-0.000693977\pi\)
\(710\) −698.502 + 365.029i −0.983806 + 0.514125i
\(711\) 0 0
\(712\) −43.6887 + 344.531i −0.0613606 + 0.483892i
\(713\) 710.943i 0.997115i
\(714\) 0 0
\(715\) 87.4066i 0.122247i
\(716\) 201.219 289.322i 0.281031 0.404081i
\(717\) 0 0
\(718\) −664.872 + 347.454i −0.926005 + 0.483919i
\(719\) 1103.79i 1.53517i −0.640947 0.767585i \(-0.721458\pi\)
0.640947 0.767585i \(-0.278542\pi\)
\(720\) 0 0
\(721\) 546.524 0.758008
\(722\) −331.179 633.729i −0.458697 0.877742i
\(723\) 0 0
\(724\) −661.925 460.357i −0.914261 0.635852i
\(725\) 203.928 0.281280
\(726\) 0 0
\(727\) −788.051 −1.08398 −0.541989 0.840386i \(-0.682328\pi\)
−0.541989 + 0.840386i \(0.682328\pi\)
\(728\) 111.385 878.387i 0.153001 1.20658i
\(729\) 0 0
\(730\) 43.6447 + 83.5165i 0.0597873 + 0.114406i
\(731\) −1611.45 −2.20445
\(732\) 0 0
\(733\) 678.996i 0.926325i −0.886273 0.463162i \(-0.846715\pi\)
0.886273 0.463162i \(-0.153285\pi\)
\(734\) −340.735 652.013i −0.464216 0.888302i
\(735\) 0 0
\(736\) 721.795 + 650.930i 0.980699 + 0.884416i
\(737\) 168.931i 0.229215i
\(738\) 0 0
\(739\) 789.744i 1.06867i −0.845274 0.534333i \(-0.820563\pi\)
0.845274 0.534333i \(-0.179437\pi\)
\(740\) −442.683 307.878i −0.598221 0.416052i
\(741\) 0 0
\(742\) −657.795 1258.73i −0.886516 1.69640i
\(743\) 216.641i 0.291576i −0.989316 0.145788i \(-0.953428\pi\)
0.989316 0.145788i \(-0.0465717\pi\)
\(744\) 0 0
\(745\) −540.234 −0.725147
\(746\) 438.048 228.919i 0.587195 0.306861i
\(747\) 0 0
\(748\) 107.033 + 74.4395i 0.143092 + 0.0995181i
\(749\) −869.399 −1.16075
\(750\) 0 0
\(751\) 932.681 1.24192 0.620959 0.783843i \(-0.286743\pi\)
0.620959 + 0.783843i \(0.286743\pi\)
\(752\) −424.049 + 157.400i −0.563894 + 0.209309i
\(753\) 0 0
\(754\) −836.286 + 437.033i −1.10913 + 0.579619i
\(755\) 1002.83 1.32825
\(756\) 0 0
\(757\) 361.451i 0.477478i 0.971084 + 0.238739i \(0.0767340\pi\)
−0.971084 + 0.238739i \(0.923266\pi\)
\(758\) 712.170 372.171i 0.939538 0.490991i
\(759\) 0 0
\(760\) 10.1282 79.8718i 0.0133266 0.105094i
\(761\) 883.718i 1.16126i 0.814168 + 0.580629i \(0.197193\pi\)
−0.814168 + 0.580629i \(0.802807\pi\)
\(762\) 0 0
\(763\) 1540.08i 2.01845i
\(764\) −92.2378 64.1498i −0.120730 0.0839657i
\(765\) 0 0
\(766\) 566.286 295.934i 0.739276 0.386337i
\(767\) 720.825i 0.939798i
\(768\) 0 0
\(769\) −177.103 −0.230302 −0.115151 0.993348i \(-0.536735\pi\)
−0.115151 + 0.993348i \(0.536735\pi\)
\(770\) 97.8908 + 187.319i 0.127131 + 0.243272i
\(771\) 0 0
\(772\) −606.969 + 872.731i −0.786229 + 1.13048i
\(773\) −327.067 −0.423114 −0.211557 0.977366i \(-0.567853\pi\)
−0.211557 + 0.977366i \(0.567853\pi\)
\(774\) 0 0
\(775\) 96.7986 0.124901
\(776\) −690.064 87.5045i −0.889257 0.112763i
\(777\) 0 0
\(778\) −198.047 378.974i −0.254560 0.487114i
\(779\) −128.273 −0.164663
\(780\) 0 0
\(781\) 123.560i 0.158208i
\(782\) 541.830 + 1036.82i 0.692877 + 1.32586i
\(783\) 0 0
\(784\) −472.220 1272.20i −0.602321 1.62270i
\(785\) 1088.05i 1.38605i
\(786\) 0 0
\(787\) 884.033i 1.12329i 0.827377 + 0.561647i \(0.189832\pi\)
−0.827377 + 0.561647i \(0.810168\pi\)
\(788\) 63.9327 91.9257i 0.0811329 0.116657i
\(789\) 0 0
\(790\) −19.1941 36.7289i −0.0242963 0.0464923i
\(791\) 1042.95i 1.31852i
\(792\) 0 0
\(793\) −45.0000 −0.0567465
\(794\) −746.485 + 390.104i −0.940157 + 0.491315i
\(795\) 0 0
\(796\) −16.7820 + 24.1300i −0.0210829 + 0.0303141i
\(797\) 1262.52 1.58409 0.792043 0.610466i \(-0.209018\pi\)
0.792043 + 0.610466i \(0.209018\pi\)
\(798\) 0 0
\(799\) −544.417 −0.681373
\(800\) −88.6276 + 98.2762i −0.110785 + 0.122845i
\(801\) 0 0
\(802\) −360.000 + 188.132i −0.448878 + 0.234578i
\(803\) 14.7735 0.0183979
\(804\) 0 0
\(805\) 1896.52i 2.35593i
\(806\) −396.961 + 207.447i −0.492507 + 0.257378i
\(807\) 0 0
\(808\) 408.051 + 51.7435i 0.505014 + 0.0640390i
\(809\) 1136.54i 1.40487i 0.711750 + 0.702433i \(0.247903\pi\)
−0.711750 + 0.702433i \(0.752097\pi\)
\(810\) 0 0
\(811\) 245.509i 0.302724i 0.988478 + 0.151362i \(0.0483659\pi\)
−0.988478 + 0.151362i \(0.951634\pi\)
\(812\) −1302.77 + 1873.19i −1.60440 + 2.30689i
\(813\) 0 0
\(814\) −74.9231 + 39.1539i −0.0920431 + 0.0481006i
\(815\) 107.223i 0.131562i
\(816\) 0 0
\(817\) 156.015 0.190960
\(818\) 576.415 + 1103.00i 0.704664 + 1.34841i
\(819\) 0 0
\(820\) 1219.49 + 848.132i 1.48718 + 1.03431i
\(821\) 733.724 0.893695 0.446848 0.894610i \(-0.352547\pi\)
0.446848 + 0.894610i \(0.352547\pi\)
\(822\) 0 0
\(823\) 1289.78 1.56717 0.783584 0.621285i \(-0.213389\pi\)
0.783584 + 0.621285i \(0.213389\pi\)
\(824\) 374.961 + 47.5474i 0.455049 + 0.0577032i
\(825\) 0 0
\(826\) 807.286 + 1544.78i 0.977343 + 1.87020i
\(827\) −468.729 −0.566782 −0.283391 0.959004i \(-0.591459\pi\)
−0.283391 + 0.959004i \(0.591459\pi\)
\(828\) 0 0
\(829\) 1356.65i 1.63649i −0.574868 0.818246i \(-0.694947\pi\)
0.574868 0.818246i \(-0.305053\pi\)
\(830\) 185.263 + 354.510i 0.223208 + 0.427121i
\(831\) 0 0
\(832\) 152.839 592.956i 0.183701 0.712687i
\(833\) 1633.32i 1.96077i
\(834\) 0 0
\(835\) 120.491i 0.144300i
\(836\) −10.3625 7.20695i −0.0123954 0.00862076i
\(837\) 0 0
\(838\) 251.399 + 481.066i 0.299999 + 0.574064i
\(839\) 1196.01i 1.42552i 0.701408 + 0.712760i \(0.252555\pi\)
−0.701408 + 0.712760i \(0.747445\pi\)
\(840\) 0 0
\(841\) 1590.59 1.89131
\(842\) 1081.93 565.405i 1.28495 0.671502i
\(843\) 0 0
\(844\) 461.969 + 321.291i 0.547356 + 0.380677i
\(845\) −418.097 −0.494789
\(846\) 0 0
\(847\) −1366.56 −1.61342
\(848\) −341.793 920.818i −0.403058 1.08587i
\(849\) 0 0
\(850\) −141.168 + 73.7730i −0.166081 + 0.0867917i
\(851\) −758.561 −0.891376
\(852\) 0 0
\(853\) 310.242i 0.363706i −0.983326 0.181853i \(-0.941790\pi\)
0.983326 0.181853i \(-0.0582096\pi\)
\(854\) −96.4385 + 50.3976i −0.112926 + 0.0590136i
\(855\) 0 0
\(856\) −596.480 75.6374i −0.696822 0.0883615i
\(857\) 1253.46i 1.46261i 0.682048 + 0.731307i \(0.261089\pi\)
−0.682048 + 0.731307i \(0.738911\pi\)
\(858\) 0 0
\(859\) 979.740i 1.14056i 0.821451 + 0.570279i \(0.193165\pi\)
−0.821451 + 0.570279i \(0.806835\pi\)
\(860\) −1483.23 1031.56i −1.72469 1.19949i
\(861\) 0 0
\(862\) −537.582 + 280.934i −0.623645 + 0.325910i
\(863\) 1240.83i 1.43781i −0.695111 0.718903i \(-0.744645\pi\)
0.695111 0.718903i \(-0.255355\pi\)
\(864\) 0 0
\(865\) 1416.78 1.63789
\(866\) −208.105 398.220i −0.240306 0.459839i
\(867\) 0 0
\(868\) −618.388 + 889.150i −0.712429 + 1.02437i
\(869\) −6.49711 −0.00747654
\(870\) 0 0
\(871\) 954.989 1.09643
\(872\) 133.987 1056.62i 0.153654 1.21172i
\(873\) 0 0
\(874\) −52.4579 100.381i −0.0600205 0.114852i
\(875\) 1302.77 1.48888
\(876\) 0 0
\(877\) 320.586i 0.365549i −0.983155 0.182774i \(-0.941492\pi\)
0.983155 0.182774i \(-0.0585078\pi\)
\(878\) 767.850 + 1469.32i 0.874545 + 1.67349i
\(879\) 0 0
\(880\) 50.8645 + 137.033i 0.0578005 + 0.155719i
\(881\) 208.356i 0.236500i 0.992984 + 0.118250i \(0.0377284\pi\)
−0.992984 + 0.118250i \(0.962272\pi\)
\(882\) 0 0
\(883\) 624.590i 0.707350i −0.935368 0.353675i \(-0.884932\pi\)
0.935368 0.353675i \(-0.115068\pi\)
\(884\) 420.816 605.070i 0.476036 0.684468i
\(885\) 0 0
\(886\) −265.590 508.220i −0.299763 0.573611i
\(887\) 1468.63i 1.65573i 0.560926 + 0.827866i \(0.310445\pi\)
−0.560926 + 0.827866i \(0.689555\pi\)
\(888\) 0 0
\(889\) −1288.13 −1.44897
\(890\) −415.349 + 217.056i −0.466684 + 0.243883i
\(891\) 0 0
\(892\) 1.35529 1.94870i 0.00151938 0.00218464i
\(893\) 52.7084 0.0590240
\(894\) 0 0
\(895\) 475.560 0.531352
\(896\) −336.534 1441.92i −0.375596 1.60929i
\(897\) 0 0
\(898\) 349.066 182.418i 0.388715 0.203138i
\(899\) 1154.21 1.28388
\(900\) 0 0
\(901\) 1182.20i 1.31209i
\(902\) 206.395 107.860i 0.228819 0.119578i
\(903\) 0 0
\(904\) 90.7362 715.549i 0.100372 0.791537i
\(905\) 1088.01i 1.20222i
\(906\) 0 0
\(907\) 418.333i 0.461227i −0.973045 0.230614i \(-0.925927\pi\)
0.973045 0.230614i \(-0.0740733\pi\)
\(908\) −870.261 + 1251.31i −0.958437 + 1.37809i
\(909\) 0 0
\(910\) 1058.94 553.388i 1.16367 0.608119i
\(911\) 69.9928i 0.0768307i −0.999262 0.0384154i \(-0.987769\pi\)
0.999262 0.0384154i \(-0.0122310\pi\)
\(912\) 0 0
\(913\) 62.7106 0.0686863
\(914\) −6.51471 12.4662i −0.00712769 0.0136392i
\(915\) 0 0
\(916\) −418.557 291.099i −0.456940 0.317793i
\(917\) −1069.94 −1.16678
\(918\) 0 0
\(919\) 1272.39 1.38454 0.692268 0.721641i \(-0.256612\pi\)
0.692268 + 0.721641i \(0.256612\pi\)
\(920\) −164.997 + 1301.17i −0.179344 + 1.41432i
\(921\) 0 0
\(922\) −201.271 385.143i −0.218298 0.417725i
\(923\) 698.502 0.756774
\(924\) 0 0
\(925\) 103.282i 0.111656i
\(926\) 498.676 + 954.243i 0.538527 + 1.03050i
\(927\) 0 0
\(928\) −1056.78 + 1171.82i −1.13877 + 1.26274i
\(929\) 675.361i 0.726977i −0.931599 0.363488i \(-0.881586\pi\)
0.931599 0.363488i \(-0.118414\pi\)
\(930\) 0 0
\(931\) 158.132i 0.169851i
\(932\) 1402.72 + 975.566i 1.50506 + 1.04674i
\(933\) 0 0
\(934\) 567.494 + 1085.93i 0.607596 + 1.16267i
\(935\) 175.931i 0.188161i
\(936\) 0 0
\(937\) −1790.67 −1.91107 −0.955534 0.294882i \(-0.904720\pi\)
−0.955534 + 0.294882i \(0.904720\pi\)
\(938\) 2046.62 1069.54i 2.18189 1.14023i
\(939\) 0 0
\(940\) −501.099 348.505i −0.533084 0.370750i
\(941\) −399.697 −0.424758 −0.212379 0.977187i \(-0.568121\pi\)
−0.212379 + 0.977187i \(0.568121\pi\)
\(942\) 0 0
\(943\) 2089.66 2.21597
\(944\) 419.469 + 1130.08i 0.444353 + 1.19712i
\(945\) 0 0
\(946\) −251.033 + 131.187i −0.265363 + 0.138675i
\(947\) −88.0644 −0.0929931 −0.0464965 0.998918i \(-0.514806\pi\)
−0.0464965 + 0.998918i \(0.514806\pi\)
\(948\) 0 0
\(949\) 83.5165i 0.0880047i
\(950\) 13.6674 7.14242i 0.0143867 0.00751834i
\(951\) 0 0
\(952\) 224.194 1768.00i 0.235498 1.85715i
\(953\) 1455.87i 1.52767i 0.645412 + 0.763835i \(0.276686\pi\)
−0.645412 + 0.763835i \(0.723314\pi\)
\(954\) 0 0
\(955\) 151.612i 0.158756i
\(956\) 625.496 + 435.022i 0.654285 + 0.455044i
\(957\) 0 0
\(958\) 183.729 96.0146i 0.191784 0.100224i
\(959\) 2306.56i 2.40518i
\(960\) 0 0
\(961\) −413.132 −0.429898
\(962\) 221.342 + 423.549i 0.230085 + 0.440280i
\(963\) 0 0
\(964\) 6.34974 9.12999i 0.00658687 0.00947094i
\(965\) −1434.51 −1.48654
\(966\) 0 0
\(967\) −1118.86 −1.15704 −0.578520 0.815668i \(-0.696369\pi\)
−0.578520 + 0.815668i \(0.696369\pi\)
\(968\) −937.576 118.891i −0.968571 0.122821i
\(969\) 0 0
\(970\) −434.744 831.905i −0.448189 0.857634i
\(971\) 1097.06 1.12983 0.564915 0.825149i \(-0.308909\pi\)
0.564915 + 0.825149i \(0.308909\pi\)
\(972\) 0 0
\(973\) 1150.89i 1.18282i
\(974\) −151.468 289.841i −0.155511 0.297579i
\(975\) 0 0
\(976\) −70.5494 + 26.1868i −0.0722842 + 0.0268308i
\(977\) 590.176i 0.604069i 0.953297 + 0.302035i \(0.0976658\pi\)
−0.953297 + 0.302035i \(0.902334\pi\)
\(978\) 0 0
\(979\) 73.4724i 0.0750485i
\(980\) 1045.56 1503.36i 1.06690 1.53404i
\(981\) 0 0
\(982\) 713.765 + 1365.83i 0.726849 + 1.39086i
\(983\) 1408.99i 1.43336i 0.697403 + 0.716679i \(0.254339\pi\)
−0.697403 + 0.716679i \(0.745661\pi\)
\(984\) 0 0
\(985\) 151.099 0.153400
\(986\) −1683.26 + 879.653i −1.70716 + 0.892143i
\(987\) 0 0
\(988\) −40.7418 + 58.5806i −0.0412366 + 0.0592921i
\(989\) −2541.59 −2.56986
\(990\) 0 0
\(991\) −939.516 −0.948049 −0.474024 0.880512i \(-0.657199\pi\)
−0.474024 + 0.880512i \(0.657199\pi\)
\(992\) −501.621 + 556.231i −0.505667 + 0.560717i
\(993\) 0 0
\(994\) 1496.94 782.286i 1.50598 0.787008i
\(995\) −39.6626 −0.0398619
\(996\) 0 0
\(997\) 1927.67i 1.93347i 0.255779 + 0.966735i \(0.417668\pi\)
−0.255779 + 0.966735i \(0.582332\pi\)
\(998\) 611.978 319.813i 0.613205 0.320454i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 216.3.h.f.53.3 8
3.2 odd 2 inner 216.3.h.f.53.6 yes 8
4.3 odd 2 864.3.h.e.593.1 8
8.3 odd 2 864.3.h.e.593.8 8
8.5 even 2 inner 216.3.h.f.53.5 yes 8
12.11 even 2 864.3.h.e.593.7 8
24.5 odd 2 inner 216.3.h.f.53.4 yes 8
24.11 even 2 864.3.h.e.593.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.3.h.f.53.3 8 1.1 even 1 trivial
216.3.h.f.53.4 yes 8 24.5 odd 2 inner
216.3.h.f.53.5 yes 8 8.5 even 2 inner
216.3.h.f.53.6 yes 8 3.2 odd 2 inner
864.3.h.e.593.1 8 4.3 odd 2
864.3.h.e.593.2 8 24.11 even 2
864.3.h.e.593.7 8 12.11 even 2
864.3.h.e.593.8 8 8.3 odd 2